Question: Perform the row operation, $3R_3\rightarrow R_3$, on the following matrix. $\left[\begin{array} {ccc} 0 & -5 & 2 & 4 \\ 1 & 6 & 5 & 3 \\ -6 & -1 & 2 & 3 \end{array} \right] $
Solution: Background There are three basic row operations that can be performed on matrices. $R_i \leftrightarrow R_j$. This symbol tells us to interchange rows $i$ and $j$. $cR_i \rightarrow R_i$. This symbol tells us to multiply a row $i$ by a constant $c$. $R_i + cR_j \rightarrow R_i$. This symbol tells us to add $c$ times row $j$ to row $i$. Finding the new row to be used For the given matrix, $R_3$ is given below. $R_3=\left[\begin{array} {ccc} -6 & -1 & 2 & 3 \end{array} \right]$ We are asked to perform the row operation, $3R_3\rightarrow R_3$. Therefore, we must multiply $R_3$ by $3$. $\begin{aligned}3R_3 &= 3\left[\begin{array} {ccc} -6 & -1 & 2 & 3 \end{array} \right] \\\\&=\left[\begin{array} {ccc} -18 & -3 & 6 & 9 \end{array} \right]\end{aligned}$ Substituting the row Now, we must substitute row $R_3$ with $3R_3$. $\left[\begin{array} {ccc} 0 & -5 & 2 & 4 \\ 1 & 6 & 5 & 3 \\ {-6} & {-1} & {2} & {3} \end{array} \right] \xrightarrow{3R_3\rightarrow R_3} \left[\begin{array} {ccc} 0 & -5 & 2 & 4 \\ 1 & 6 & 5 & 3 \\ {-18} & {-3} & {6} & {9} \end{array} \right] $ Summary Our resultant matrix is the following. $\left[\begin{array} {ccc} 0 & -5 & 2 & 4 \\ 1 & 6 & 5 & 3 \\ -18 & -3 & 6 & 9 \end{array} \right]$